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''of a category $\mathcal{C}$ having a terminal object''

A pair $(X,x_0)$ where $X \in \mathrm{Ob}\,\mathcal{C}$ and $x_0$ is a morphism of the [[terminal object]] into $X$.  Examples are [[pointed set]]s, and pointed topological spaces (see [[Pointed space]]). The pointed objects of $\mathcal{C}$ form a category, in which the morphisms are the mappings sending the distinguished point to the distinguished point.


====Comments====
The category of pointed objects of $\mathcal{C}$ has a zero object (see [[Null object of a category|Null object of a category]]), namely the terminal object of $\mathcal{C}$ equipped with its unique point. Conversely, if a category $\mathcal{C}$ has a zero object, then it is isomorphic to its own category of pointed objects.